M[any] Vacua of Iib

Description of the spectrum of fluctuations around a commutative vacuum solution, as well as around a solution with degenerate commutator in IIB matrix model is given in terms of supersymmetric Yang–Mills (YM) model. We construct explicitly the map and study the dependence of the spectrum and respective YM model on the symmetries of the solution. Introduction IKKT, or IIB matrix model [1], is a statistical mechanic model which was introduced as a nonperturbative regularisation of IIB superstring model in the Schild gauge [2]. It is the string analog of lattice gauge models and provides a tool for non-perturbative numerical study of the string theory (M-theory) [3, 4, 5, 6]. The picture one has in the IKKTmodel is closely related to the Connes’ approach to the noncommutative geometry [7]. In this approach a manifold is described in terms of the so called Connes’ triples (A,∆,H), rather than as a set of points. In this triple, A is the algebra of bounded operators (algebra of functions), ∆ is some elliptic differential operator (Laplace or Dirac operator) and H is the Hilbert space where A and ∆ are represented. “Points” of such a manifold can be identified with the spectra of some position operators built from the triple, other local and global characteristics can as well be extracted from it. In the case of IKKT matrix model the rôle of A is played by “smooth” Hermitian matrices with bounded square trace in the limit N → ∞, while the ∆ and H are represented by the background solution and the adjoint representation of U(N). Compactifications of this model to d-dimensional noncommutative tori was shown to result in noncommutative Yang–Mills (YM) models in respective dimensions [8]. In particular, compactification to a circle yields the Banks–Fischler–Shenker– Susskind (BFSS) matrix model which was introduced as a non-perturbative regularisation of the light-cone membrane action in D = 12, [9]. The compactification in Ref. [8] is introduced as a restriction on the background solution to satisfy some periodicity conditions modulo gauge transformations. The respective periods are identified with the periods of the torus on which the model is compactified. From the other hand, given a BPS background, i.e. a solution to equations of motion which preserves a part of the supersymmetry, one can map the space of N × N Hermitian matrices of IKKT model to the space of real or matrix valued functions on some non-commutative manifold. Under certain conditions this map is isomorphism of algebras where the matrix product is mapped into noncommutative or star product of functions due to the noncommutativity of the manifold. The properties of the manifold are exclusively determined by the respective BPS background solution. Explicitly, a BPS solution is given by a set of matrices having scalar commutators. In the case when the matrix of commutators is nondegenerate (or for the subset on which it is nondegenerate) one can easily construct the map Work supported by RFBR grant # 99-01-00190, INTAS grant # 950681, and Scientific School support grant 96-15-0628. 1 2 CORNELIU SOCHICHIU from the space of arbitrary Hermitian matrix fluctuations around given BPS background to the space of functions on a noncommutative manifold [6]. The limit of commutative manifold corresponds to infinite commutator of the solution. The case when the commutator is degenerate corresponds just to an opposite situation. One may argue, and this was the main reason for arriving at actual paper, that only commutative solutions give the global minimum of the IKKT actions. However, as we show in the Section 2., this is not exactly the case, since due to homogeneity of the IKKT action in bosonic fields, the respective action vanishes on purely bosonic solutions. Unfortunately, this regretful error committed in the pioneering work [1], was reproduced in the succeeding papers. In spite of this disappointing discovery the study of the spectrum of fluctuations around a commutative solution still presents some interest, in particular due to the fact that, as it will be seen below, the degeneracy in the commutator of the solution corresponds to increasing the dimensionality of corresponding noncommutative YM model, which gives a “physically” different model (compare this with the situation with the first and second class constraints, [11]). To obtain a noncommutative U(1) YM model from the matrix fluctuations one has to impose some irreducibility condition, while allowing certain degeneracy to the background solution one comes to description of a “vector fibre bundle” over the noncommutative manifold which leads to “nonabelian” noncommutative YM. Respective degeneracy can be interpreted as compactification on many coinciding branes, while the nondegenerate case corresponds to a single brane [1]. The objective of the actual paper is to analyse the relation between the background commutative/degenerate vacuum solution and the properties of the resulting YM model. Although, the commutative solution is a particular case of a generic BPS solution, as we already mentioned, it is “physically” different, since it corresponds to a singular limit of the generic case. The plan of the paper is as follows. In the next section we give a brief account of IIB matrix model. After that we consider a BPS solution, show that it corresponds to a vanishing action and consider in more details the commutative case. We impose a set of conditions such a solution must respect, and build explicitly the map between the matrix fluctuations and functions of noncommutative manifold which appears to be a “noncommutative” product of two commutative manifolds dual to each other. This allow to find in the Section 4. a representation of IIB matrix model in terms of YM fields for both commutative and generic degenerate case. Finally, we discuss the results and consider the consequences and possible generalisations of the actual analysis. 1. The IIB Matrix Model The IKKT, or IIB matrix model, is described by the classical action: